We also show that every theory with \(\mathrm {SOP}_3\)  (or with \(\mathrm {SOP}_n\)  for \(n>3\) ) admits types that mimic arbitrary distributions of types corresponding to (pre-)cuts in ultrapowers of infinite linear orders. The idea is similar to Shelah's proof in [1]} that theories with \(\mathrm {SOP}_3\)  are maximal in the Keisler order, but avoids use of the bi-interpretability ordering defined in the same paper.
